Deciding some Maltsev conditions in finite idempotent algebras

In this paper we investigate the computational complexity of deciding if a given finite algebraic structure satisfies a certain type of existential condition on its set of term operations. These conditions, known as Maltsev conditions, have played a central role in the classification, study, and applications of general algebraic structures. Several well studied properties of equationally defined classes of algebras (also known as varieties of algebras), such as congruence permutability, distributivity, and modularity are equivalent to particular Maltsev conditions. The general set up for the decision problems that we consider in this paper is as follows: for a fixed Maltsev condition Σ, an instance of Σ-testing is a finite algebraic structure (or just algebra for short) A. The question to decide is if the variety of algebras generated by A satisfies the Maltsev condition Σ. This is a natural computational problem in universal algebra (especially in the case of important Maltsev conditions such as having a majority operation) that finds practical applications in the UACalc [7] software system for studying algebras on a computer. Moreover, some Maltsev conditions are also known to correspond to complexity classes of the Constraint Satisfaction Problem (see the recent survey [2] for an overview) and deciding these Maltsev conditions for algebras of polymorphisms of relational structures is an important meta-problem (which, however, is beyond the scope of this paper). It turns out that deciding many common Maltsev conditions for finite algebras is often EXPTIME-complete [8, 13] and so some of the recent work